Solution Quality

We evaluate the solution quality based on the relative gap between the passenger in- convenienceP I in the solution and a lower bound LB. This lower bound is calculated

about the disruption length. TheP I and LB depend on the disruption scenario δ. The

relative gap, minimized in the objective function, is calculated as r =



δ∈D_(P Iδ−LBδ) δ∈DLBδ , whereP Iδ andLBδ represent the passenger inconvenience and lower bound in scenario

δ, respectively. Moreover we report the relative gap per scenario rδ = P Iδ_LB−LBδ

δ . TheLB represents absolutely unavoidable inconvenience. However, because of uncertainty in the disruption duration and limited available capacity, it is likely that there does not exist a solution with a 0% relative gap. Note that the gap does not represent an optimality gap, but provides insight into the current quality of the solution, and an upper bound on the maximum potential for improvement of the solution.

Settings Relative Gap (%) Mean Delay (min) Delayed Passengers (Nr)

Case Advice Total S M L S M L S M L

D1 UA 8.33 11.22 6.49 7.67 34.4 33.7 34.2 12942 14217 15342 D1 SPA 8.23 11.21 6.49 7.40 34.4 33.7 34.2 12945 14217 15325 D1 OPA 7.92 10.19 6.48 7.39 34.0 33.7 34.2 12949 14216 15324 D2 UA 35.83 45.16 32.26 31.54 53.8 53.4 54.4 26196 27214 28899 D2 SPA 33.63 36.03 31.33 33.78 51.9 52.4 54.9 25436 27534 29127 D2 OPA 14.31 14.57 13.45 14.91 48.4 50.0 52.0 22961 24936 26373 D3 UA 6.55 12.84 2.92 4.95 47.2 47.0 48.5 9223 9862 10831 D3 SPA 5.74 10.70 2.92 4.45 47.6 47.0 48.4 8976 9862 10792 D3 OPA 5.25 11.16 2.53 3.15 47.9 46.8 47.8 8954 9865 10795 D4 UA 8.86 14.16 6.36 6.88 41.9 42.3 43.3 17486 18358 19611 D4 SPA 8.58 13.53 6.36 6.63 42.1 42.3 43.2 17290 18358 19593 D4 OPA 6.13 10.44 3.89 4.71 42.0 42.3 43.3 16856 17967 19202 D5 UA 88.90 86.21 88.21 91.74 46.0 48.4 51.3 20496 22100 23117 D5 SPA 94.37 94.18 98.16 91.04 48.2 50.0 50.9 20397 22531 23191 D5 OPA 8.90 11.27 9.89 6.06 34.9 34.9 34.8 16128 17932 18854

Table 5.1: Summary of solution quality for UA, SPA, and OPA

Table 5.1 presents the relative overall gap, and relative gap per disruption duration scenario for all cases (D1 to D5), and for each advice optimization model (UA, SPA and

OPA. The smaller the gap, the smaller the delay resulting from uncertainty and capacity

constraints. A lower relative gap within a column per case indicates a better solution. Comparing relative gaps between cases and columns only indicates the difference in delay resulting from the uncertain duration and the limited available capacity. However, here a higher relative gap does not necessarily indicate a worse solution, as part of the delay due to capacity constraints and uncertainty may be unavoidable. Furthermore, Table 5.1 includes the average delay per affected passenger per scenario, and the total number of passengers affected by the disruption per scenario. A longer disruption duration doe snot necessarily lead to longer average delays, but does increase the number of affected

5.8 Computational Results 129

passengers. Results are presented for the best settings ofδI

qandδrIper case and per advice optimization. The table thus allows comparison of the relative and absolute impact of the proposed method on passenger inconvenience.

The results in Table 5.1 show that OPA provides consistently better solutions than

SPA and UA, and generally SPA performs slightly better than UA. Especially in case of

large inconvenience, as in scenarios D2 and D5, OPA solution provides solutions with a much lower passenger inconvenience than SPA and UA. In case of D2, the OPA solution significantly reduces inconvenience, in comparison to both SPA and UA, while SPA is slightly better than UA. It reduces the average delay of affected passengers by up to 10%, and also reduces the number of affected passengers by around 10%.

In case of D5, the OPA solution reduces delay dramatically, while SPA is slightly worse than UA. It reduces the relative gap by a factor up to 10. The number of affected passengers is reduced by 20%, while average delay of affected passengers is reduced by up to 30%. An explanation of this dramatic difference is that the advice helps passengers to avoid traveling into a bottleneck where capacity shortages lead to large delays. Indeed, the number of passengers needing to compete for seats is much lower in the OPA solution than in the SPA and UA solutions.

In cases D1, D3 and D4 the advantage of using OPA is small, and the benefit of using

SPA is even smaller. Improvements result from a small reduction in the average delay

of affected passengers, and/or a reduction in the number of affected passengers by the disruption. Kroon et al. (2014) found that the delay reduction for these cases resulting from rolling stock rescheduling is small, and therefore it is likely that not much more improvement is possible, as the relative gaps are also fairly small. We can thus conclude that the benefit of the method is dependent on the location of the disruption. A more extensive case analysis would be required to conclude whether or not the methodology provides a benefit in the majority of disruptions an operator experiences.

The results in Table 5.1 are based on the best settings for δI

q and δ

I

r selected per case and per advice optimization model. These settings represent either an optimistic or pessimistic estimate of the length of the disruption. In analyzing these settings we find that the best choice forδI

qandδ

I

rdepends both on the case and the optimization model. The performance of UA depends most on these settings, with a difference of up to 24 percent points in the relative gap. It generally performs best on medium estimates for bothδI

q andδ

I

r, where the selection ofδ

I

qinfluences the results the strongest. SPA varies the least over the selection ofδI

q andδ

I

r. For most settings, therefore, SPA outperforms

UA by a much larger margin than based on the best settings, as presented in Table 5.1.

In contrast to UA, the performance depends most strongly on the selection ofδI

0 50 100 150 200 0.94 0.96 0.98 1 Delay (minutes) P assengers (p ercen tage full p opulation)

Cumulative delay distribution D2 DisrScen S

UA SPA OPA

Figure 5.8: Cumulative Delay distribution for Case D2, scenario S

thanδI

q. However, there is no dominant setting forδ

I

rover all cases. Finally the sensitivity of the OPA solution is in between those of UA and SPA. The OPA solutions also depend strongly onδI

r, and are generally not best for a medium selection forδIqandδIr. There is however no dominant strategy, e.g. to be pessimistic or optimistic, in the selection ofδI

q andδI

r. Therefore, currently the best performance is obtained by analyzing all values for

δI

qandδrI.

That OPA solutions perform better by reducing the number of affected passengers, the average delay per passenger, and the worst case delay, can also by deduced from the cumulative delay distribution represented in Figure 5.8. This graph represents the cumulative delay distribution of solutions UA, SPA and OPA for the short scenario and case D2, with the horizontal axis reflecting the delay per passenger, and the vertical axis the percentage of passengers that have experienced at least this amount of delay. Results for other cases and scenarios are similar. The OPA solution has a higher percentage of passengers with zero minutes delay, and the left-shift in comparison to UA and SPA shows that it also has fewer passengers with long delays.